- #176
PAllen
Science Advisor
- 9,209
- 2,430
Reading the argument as presented in MTW a bit generously, I think the following can be said:PeterDonis said:In the argument as presented in MTW, the word "geodesic" is not mentioned at all. Without being able to see Schild's actual papers, I can't tell whether he used that term himself or not.
Even if he did, as I pointed out in response to PAllen, he must be using the term in a different sense from its usual one; if the point is that the timelike sides of the quadrilateral he describes must be geodesics in order for it to be properly termed a "parallelogram", then they can only be geodesics of the background flat Minkowski metric, which is not physically observable locally--that is to say, since these worldlines have nonzero proper acceleration, they are obviously not geodesics in the usual sense, so any definition of them as geodesics must be relying on some non-local measurement (such as exchanging light signals with observers who are very far away).
1) They posit a theory where the Minkowski metric is the observable metric for distance and time measurements.This is pretty clearly stated.
2) They also state free fall paths and light paths near a gravitating body are not geodesics. Gravity is governed by a field of unspecified nature that does not change observable geometry.
3) They posit it is possible to set up global Lorentz frame physically using a described procedure.
4) Though not clearly stated, the implication is that straight lines in the global Lorentz frame are Minkowski geodesics. I don't see any other reasonable way to read their argument.
5) The figure they set up has two parallel straight (geodesic) timelike sides and two congruent, possibly curved sides that need not be geometrically parallel.
6) They then note that allowing gravitational time dilation leads to a contradiction.
Thus the assumptions must be changed. With a fair amount of unstated reasoning, I think you could get to the conclusion that the observable geometry must be a curved pseudoriemannian manifold. Of course, I think there are other routes to this conclusion that are much more straightfowrd.
The interesting thing isn't the claimed proof of curvature (which is incomplete as given), but the concnclusion that gravity as a field theory of SR cannot accommodate gravitational time dilation between static observers (as long as the Minkowsi metric remains the observable metric).
[edit: a lot rests on the argumentation that the static observers would have to be straight lines in minkowski geometry if gravity is a field theory on SR (with SR geometry being the observable geometry). The argument stands or falls on how well the case for this is made.]
Last edited: